Molecular dynamics investigation of thickness effect on liquid films

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 14 8 OCTOBER 2000 Molecular dynamics investigation of thicness effect on liquid films Jian-Gang Weng, Seungho Par, a) Jennifer R. Lues, and Chang-Lin Tien b) Department of Mechanical Engineering, University of California, Bereley, California Received 19 April 2000; accepted 14 July 2000 This wor applies the molecular dynamics simulation method to study a Lennard-Jones liquid thin film suspended in the vapor and to explore the film thicness effect on its stability. For the accurate estimation of local pressure distributions in the film, an improved method is proposed and used. Simulation results indicate that profiles of the local surface tension distribution vary widely with film thicness, while surface tension values and density profiles show little variation. As the film gets thinner, the two liquid vapor interfacial regions begin to overlap and liquid-phase molecules in the center region of the film experience larger tension in the direction parallel to the film surface. Such interface overlapping is believed to destabilize the film and the occurrence of film rupture depends on the system temperature and the cross-sectional area of the computational domain American Institute of Physics. S I. INTRODUCTION The liquid vapor interface has been subject to extensive study for more than one century because of its critical importance in many industrial applications such as phasechange heat transfer, spread wetting, and material processing. The thicness of an interfacial region is in the nanometer range, maing experimental studies of such a thin region extremely difficult. Although there exist a few experimental wors on the liquid vapor interface, 1 physical understanding of interfacial phenomena still relies heavily on theoretical analysis and numerical simulations. Molecular Dynamics MD simulation is one of the most effective tools to study interfacial phenomena since it can yield detailed information on the molecular structure of an interface if the appropriate intermolecular potential is given. The past 10 years have seen a number of reported MD simulations on three-dimensional planar liquid vapor interfaces after the pioneering wor by Chapela et al. 2 Specifically, Nmeer et al. 3 and Daiguji and Hihara 4 calculated the local surface tension of a liquid film sandwiched by its vapor. Mece et al. 5 investigated the influence of the cut-off radius on interfacial properties and proposed a new longrange force correction method. Hwang et al. 6 applied MD simulations to observe the transient evolution of rupture processes of a free liquid film and a film on a solid substrate. Despite the increasing practical importance of thin liquid films, however, the effect of liquid film thicness on interfacial properties, such as density, surface tension, and local pressures, has not been reported. The object of this wor, therefore, is to investigate the film thicness effect on the liquid vapor interface. The interfacial property simulation techniques are very similar to those adopted in the previous MD studies 3 5 except that an improved method is developed in order to obtain a better a Also at Department of Mechanical Engineering, Hongi University, Seoul , Korea. b Author to whom correspondence should be addressed. estimation of the local surface tension profile across the film. The film stability analysis then follows in which results from classical thermodynamic theory and MD simulation are qualitatively compared. Both results indicate that the film thicness has a significant effect on film stability. II. SIMULATION TECHNIQUE This simulation uses the well-nown Lennard-Jones LJ 12-6 potential, given as r 4/r 12 /r 6. 1 For the ease of physical understanding, the LJ fluid is assumed to be argon, and parameters for argon are listed as follows: 7 the length parameter 0.34 nm, the energy parameter J, and the molecular mass m g. The cut-off radius r c beyond which the intermolecular interaction is neglected is 5.0. Separate runs with r c 6.5 are carried out and the results in both surface tension values and local stress profiles discussed in Sec. IV are very close to those with r c 5.0, which indicates that the cut-off radius at 5.0 is large enough. The long-range force correction is not used in this preliminary study since the error introduced by neglecting long-range forces is about 10% for this cut-off radius. 5 The simulation domain is schematically shown in Fig. 1, with periodic boundary conditions applied in all three directions. Simulation domain dimensions, system temperatures T*, and initial film thicnesses L s * are listed in Table I together with simulation results of equilibrated film thicness L* f and surface tension *. The equations of motion are solved by using the velocity Verlet algorithm 7 with a time step of 5 fs, or * Note that in this wor, all quantities with an asteris, such as * and *, are nondimensionalized according to,, and m. At the beginning of the simulation, a liquid sheet of initial thicness L s * is placed at the center of the computational domain and its two sides are filled with vapor molecules. The initial density of the liquid sheet is 0.8, which is /2000/113(14)/5917/7/$ American Institute of Physics

2 5918 J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 Weng et al. FIG. 2. Density profiles for L s *8.55 S1, L s *6.84 S2, L s *5.13 S3, and L s *4.70 S4 at T* FIG. 1. System configuration for a thin film in its vapor. slightly higher than the experimental value of saturated bul liquid argon e.g., at T*0.818). 8 In contrast, the vapor density, 0.008, is slightly lower than the experimental saturated vapor argon density at that temperature. 8 The vapor and liquid molecules have been equilibrated individually for time steps at the designed temperature T* before they are put together into the computational domain. The computational domain is artificially divided into many thin slabs in the direction normal to the film surface z direction with the slab thicness L sl * equal to 0.1. Timeaveraged values of the density and pressure in each slab are considered as local values. In the simulation, the equilibration period of time steps, in which velocity rescaling is performed at each step, is to mae sure that the system is at the designed temperature, followed by a relaxation period of another time steps. Then the production period of time steps starts, in which instantaneous values of the local density and stress are calculated at each time step and time-averaged values are obtained at the end of this period. III. DENSITY PROFILES AND SURFACE TENSION It is of vital importance that the system be at equilibrium before statistical values of the local density and pressure are TABLE I. Simulation conditions and some simulation results. indicates that film rupture occurs. Label Simulation conditions Film thicness L x *L y *L z * T* L s * initial L f * final S S S S S H H L L * taen. Due to current computation capacity limitations, the MD method cannot simulate a macroscopically long period. Some criteria have to be chosen to determine whether the system is at equilibrium, but unfortunately, such choices in the literature are still arbitrary. Here, the system is considered to be at equilibrium when the local temperature and normal pressure in each slab are constant. During the production period, the inetic energy of molecules in each slab is measured and the average values in each slab indicate that the temperature is almost the same throughout the system. The mechanical equilibrium requirement that the normal pressure in each slab should be uniform is also satisfied and will be discussed in the next section. Simulated density profiles in cases S1 S4 are shown in Fig. 2, where Z*0 corresponds to the center of the film. The vapor densities * v in these cases are about 0.01, and the liquid densities at the film center l * vary around 0.775, both of which are close to their counterparts as the bul saturated densities. The apparent film thicness is determined as the distance between the two equimolar dividing surfaces in the two interfacial regions. The equimolar dividing surface is defined as the surface on which the local density is 0.5(* v l *). Simulation results of film thicness are shown in Table I as well as the corresponding surface tension values. In some runs S5, H2, and L2, film rupture occurs and local density profiles and surface tension values cannot be estimated. The surface tension is calculated by the virial expression 5 1, 2 4A i j r r c r 3z 2 r r where AL x L y and the angle bracets refer to a time average. The intermolecular distance between two molecules i and j is r and its components in each direction are denoted as x, y, and z, respectively. The derivative of LJ potential with respect to r is. Two conclusions can be drawn from the comparison of surface tension values in Table I. One conclusion is that at the same temperature, the surface tension values vary only slightly with film thicness, with an average value of 0.78 at T*0.818 that is consistent with the result in the previous wor. 5 The other is that the average surface tension value is about 12% higher than the

3 J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 Thicness effects on liquid film stability 5919 experimental value about 9.9 mn/m for argon at that temperature, corresponding to 0.69 in a nondimensional system. There are several reasons for this deviation, such as neglecting many-body interaction in the LJ potential, neglecting the long-range force correction, 5 and neglecting the finite size effect of the cross-sectional area in the computational domain. 9 IV. LOCAL STRESS PROFILES Local stress also understood as local surface tension 3,4 is defined as the difference between the local normal and tangential pressure components. Nmeer et al. 3 and Daiguji and Hihara 4 calculated the local stress distribution across the thin film. In their wors, the local normal pressure component p N and tangential pressure component p T are expressed as p N,K p N,I, p N n B T 1 V sl z 2 r r p T n B T 1 V sl 1 2 x 2 y 2 r r p T,K p T,I, 4 where n() is the number density in slab, V sl is the volume of slab (V sl L x L y L sl ) and L sl is the slab thicness. The first terms in Eqs. 3 and 4, p N,K and p T,K, are the contribution from inetic motion of molecules, while the second terms, p N,I and p T,I, are the contribution from the intermolecular force. The summation runs over all particle pairs (), of which at least one of the particles is situated in slab. If only one particle is in slab, half of the intermolecular force contribution is given to slab, while the total contribution is given to slab if both molecules are in that slab. The contribution of pair () is z 2 (r )/r for p N and (x 2 y 2 )(r )/(2r ) for p T. The surface tension can be expressed as L zpn p T dz, where p N and p T refer to the local values. One can prove that Eq. 5 is just another form of Eq. 2. It should be noted that this is a simplified approach to calculate the local pressure components and is adopted mainly for computational efficiency. 3,4 If detailed information on local pressure components is needed, a more accurate method should be developed. According to Kirwood and Buff s theory, the intermolecular force contribution to the local normal pressure component is given as the sum over the normal components of all the pair forces acting across a surface element divided by the surface area. 10 The same argument is applicable for the tangential pressure component. In other words, an intermolecular force contributes to local pressure in each plane between the two molecules. Here, the intermolecular force is assumed to act in a straight line. Since in MD simulation of local pressure in a planar interface, a volumetric average is preferred, as can been seen in 3 5 FIG. 3. Schematic pressure combination of the pair. Eqs. 3 and 4, Kirwood and Buff s local pressure tensor is averaged over a distance L sl. All of these form the physical basis for an improved method to calculated local pressure profiles. The whole procedure for the normal pressure component can be expressed as p N,I 1 L sl slab_ 1 A 1 V sl slab_ F z dz 1 V sl Fz L, 1 V sl Fz z f,, F z dz where F z is the normal component of the intermolecular force between molecules i and j and the factor f, is defined as L, /z. The third step in the derivation is due to the fact that F z is constant along the line connecting molecules i and j. The length L, is defined as the size of the interval in which F z is effective in the slab. For example, consider the force between the pair in Fig. 3. If neither molecules is in a slab, such as slab 2, L 2, L sl ; if one molecule is in a slab, such as slab 1, L 1, L 1 ; and if both molecules are in a slab not shown in Fig. 3, L, z. According to this change, the pressure tensor is modified as p N n B T 1 V sl z 2 r r f,, 7 6 p T n B T 1 1 V sl 2 x 2 y 2 r r f,. 8 It should be noted that these two methods yield exactly the same value of surface tension, 3,4 since after substituting Eqs. 7 and 8 to Eq. 5 the integral of f, always gives one. In the bul liquid with uniform density, these two methods are expected to yield statistically the same results, too. 3,4

4 5920 J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 Weng et al. FIG. 4. Normal pressure components calculated from a the simplified method and b the improved method at T*0.818 and L s *8.55. However, in an interfacial region, the local stress profiles calculated by these two methods are quite different. As a demonstration, Fig. 4 shows profiles of the normal pressure component for case S1. It is found that p N * calculated from the simplified method has significant fluctuation across the interface while the fluctuation disappears in the improved method. Although such fluctuation in the simplified method might be averaged out in a simulation with enough long time, it is quite clear that the improved method can mae better estimation of the local pressure. As stated in the previous section, the requirement of mechanical equilibrium is satisfied because the normal pressure does not change significantly. Figure 5 plots profiles of local values of pressure components p N,I *, p T,I *, and stress (p N *p T *) using both simplified and improved methods. It is found that although the surface tension values show little variation with the change of film thicness, the stress distributions are quite different. When the film becomes thinner and thinner, the two interfaces start to interact with each other and the local stress at the center rises, which causes the center liquid to be under the metastable condition. V. FILM STABILITY As the film thicness decreases further S5, film rupture occurs. Figure 6 shows the initial and final position of molecules in the computational domain. From the snapshot of the final molecule position, it is obvious that the film has broen. This indicates that at the given temperature and domain size, the liquid film with the equilibrium thicness L* f less than 5.03 is unstable. Similarly, increasing the temperature T* or crosssectional area A (AL x L y ) will destabilize the film. As indicated in Table I, when T* increases to 0.85, the film that is stable at T*0.818 S3 becomes unstable H2; while as L x * and L* y increase from 17.1 to 20.26, the minimum stable film thicness changes from 5.03 S4 to 5.43 L1. It should be emphasized that although a film is stable in the simulation period of time steps corresponding to 800 ps, that duration is still too short to guarantee against the occurrence of film rupture at some later time. In this sense, what is compared in this wor is the relative stability of the film within a very short period. According to the three simulations above, the minimum thicness of the stable film depends on both the computational domain size and system temperature. Classically, surface wave theory is applied to study the stability problem of liquid films and cylinders. Many macroscopic models have been developed to study stability of a liquid film on a solid substrate 11,12 and a similar model is now proposed to study the free film stability. Consider the most unstable case in which two synchronic one-dimensional waves propagate along the x direction at each liquid vapor interface the squeezing mode 11, as shown in Fig. 7. Assume the surface wave can be expressed as a sine wave with wavelength and amplitude L. The average film thicness is L f. With the assumption that the curvature dependence of surface tension is neglected, the change in total surface energy due to this wave is W s 2 lvl 2. 9 Note the surface energy change is positive due to the increased surface area. The volumetric energy density in the film can be written as 12 W vdw A 2, 10 12L f where A is the Hamaer constant. The volumetric energy change due to this 2 wave is W v L A 2L f The volumetric energy change is negative because A is always positive for two identical media vapor in this case interacting across another medium liquid film 13. The Hamaer constant can be expressed as A BT h e n 2 1 n n 2 1 n 2 2, 12 3/2 where it is different from the energy parameter in the LJ potential is the dielectric constant, n the refractive index, h Planc s constant, v e the main electronic absorption frequency, and the subscripts 1 and 2 refer to the vapor and liquid, respectively. For the vapor phase, 1 and n 1 are close

5 J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 Thicness effects on liquid film stability 5921 FIG. 5. Profiles of inertial parts of local normal pressure p N,I *, inertial parts of local tangential pressure p T,I *, and local stress p N *p T *. The left column is from the simplified method and the right column is from the modified method. to 1, while for the liquid phase, the liquid argon properties are used and 2 and n 2 are and at T*0.818 or T99 K), respectively. 14 Typically, e is about s 1 for various materials. 13 From Eq. 12, the Hamaer constant A at T*0.818 is about J. The total energy change (W s W ) should be positive for the film to be stable, that is, the wavelength should be smaller than the critical length, cr L 2 43 l f, 13 A For L f 5.03 S4 the critical wavelength cr is Due to the periodic boundary condition assumption, L x can be understood as the longest wavelength allowed by the computation domain. According to this model, if cr L x, a film may become unstable. For the same film at a higher temperature, since the surface tension l decreases and cr becomes smaller, the film will be less stable. The simulation results in Table I show that, for a film with L f 5.03, L x 17.1 is the stable condition S4 while increasing L x to destroys the film L2. This seems to indicate that the value of the critical wavelength cr

6 5922 J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 Weng et al. FIG. 7. Perturbation of a free liquid film suspended in its vapor. is somewhere between 17.1 and There are several possible reasons for the difference between the results predicted by the classical thermodynamic model and MD simulation. First, the definition of film thicness is highly ambiguous. The classical approach assumes a zero-thicness interface and a well-defined film thicness, while MD shows that for a very thin film, the interface thicness is on the same order as the bul film thicness. Secondly, the classical model treats one dimensional wave, while in this MD simulation, a wave in the y direction also exists which may further destabilize the film. Thirdly, the main electronic absorption frequency is not an exact value. In addition to the surface wave argument, it might also be reasonable to assume that the film becomes unstable because its center cannot sustain the large tensile stress in its metastable state. For a film with the same thicness but different crosssectional area, the center liquid density and local stress distribution are the same. But the perturbation not necessarily the surface wave depends on the cross-sectional area of the computational domain. The film with the larger cross-sectional area is less stable due to the increased perturbation. Similarly, perturbation is intensified at higher temperatures, which explains why increasing temperature will destabilize the film. When rupture occurs, the center liquid will release the stress. It can be expected that if the liquid is under larger stress i.e., if the film is thinner, the rupture rate will be higher. However, the film thicness effect on the rupture rate has not been carefully investigated. The exact location where rupture initiates is also unnown. VI. CONCLUDING REMARKS FIG. 6. Snapshot of a initial position of molecules initial film thicness L s *4.28 and b final position of molecules after steps. This wor studies the film thicness effect on interfacial properties. It is found that as the film thicness decreases, the two interfacial regions begin to overlap, which increases the local stress at the center of the film. As the local stress exceeds the maximum value, film rupture occurs. An improved method is proposed and found to be able to yield a more accurate profile of the normal pressure component, which indicates that it can obtain a better estimation on local stress as well. A macroscopic thermodynamic analysis is developed to study stability of the free film. Since it is still questionable whether macroscopic theories can be directly applied to such a thin film, the results between the thermodynamic model and the MD simulation cannot be quantitatively compared. It is also important to study the film stability of a film on the solid substrate or a film between two solid plates, since these two cases are frequently encountered in industrial applications, such as spread wetting and phase-change cooling. A detailed investigation on the local stress profile and film stability may reveal some new understanding on wetting coefficients and disjoining pressures.

7 J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 Thicness effects on liquid film stability 5923 ACKNOWLEDGMENTS The authors gratefully acnowledge financial support from the U.S. Department of Energy and the National Science Foundation. Seungho Par has been partially supported for this research by grants from the Korea Science and Engineering Foundation under Contract No O. D. Velev, T. D. Gurov, I. B. Ivanov, and R. P. Borwanar, Phys. Rev. Lett. 75, G. A. Chapela, G. Saville, S. M. Thompson, and J. S. Rowlinson, J. Chem. Soc., Faraday Trans. 2 73, M. J. P. Nmeer, A. F. Baer, C. Bruin, and J. H. Sien, J. Chem. Phys. 89, H. Daiguji and E. Hihara, Heat and Mass Transfer 35, Reference 4 can be found at: /bibs/ / htm 5 M. Mece, J. Winelmann, and J. Fischer, J. Chem. Phys. 107, C.-C. Hwang, J.-Y. Hsieh, K.-H. Chang, and J.-J. Liao, Physica A 256, M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids Oxford University Press, New Yor, N. B. Vargafti, Table on the Thermophysical Properties of Liquids and Gases Hemisphere, Washington, D.C., 1975, p L.-J. Chen, J. Chem. Phys. 103, J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity Clarendon, Oxford, 1982, Chap. 4, pp D. A. Edwards, H. Brenner, and D. T. Wasan, Interfacial Transport Processes and Rheology Butterworth-Heinemann, Boston, MA, 1991, Chap. 12, pp A. Majumdar and I. Mezic, Microscale Thermophys. Eng. 2, J. Israelachvili, Intermolecular and Surface Forces Academic, London, 1992, Chap. 11, pp Numerical Data and Functional Relationships in Science and Technology, Vol. 6-II/8, Landolt Bornstein Springer, New Yor, 1962; Vol. IV/6, Landolt Bornstein Springer, New Yor, 1982.